09-05-2012, 04:53 PM
Reduced-Order Modeling of Time-Varying Systems
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INTRODUCTION
VERIFYING systems hierarchically at different levels
of abstraction is an important task in communications
design. For this task, small macromodels need to be generated
that abstract, to a given accuracy, the behavior of much bigger
subsystems. For systems with time varying and nonlinear
blocks, macromodels are typically constructed by manually
abstracting circuit operation into simpler forms, often aided
by extensive nonlinear simulations. This process has disadvantages.
Simulation does not provide parameters of interest (such
as poles and zeros) directly; obtaining them by inspection
from frequency responses can be computationally expensive.
Manual abstraction can miss nonidealities or interactions that
the designer is unaware of. Generally speaking, manual macromodeling
is heuristic, time consuming, and highly reliant on
detailed internal knowledge of the system under consideration.
PAD´E APPROXIMATION OF THE LTV TRANSFER FUNCTION
The LTV transfer function (7), (19), and (26) can be
expensive to evaluate, since the dimension of the full system
can be large. In this section, methods are presented for
approximating using quantities of much smaller
dimension.
The underlying principle is that of Pad´e approximation, i.e.,
for any of the forms of the LTV transfer function, to obtain a
smaller form of size whose first several moments match those
of the original large system. This can be achieved in two broad
ways, with correspondences in existing LTI model-reduction
methods.
MACROMODELING CYCLOSTATIONARY NOISE
When a system is macromodeled, it is also desirable to
replace all its noise contributions by a few equivalent noise
sources at the inputs or outputs.3 Usually, the power spectra
of the equivalent sources have complicated frequencydependence,
unlike those of the relatively simple white and
flicker noise models typically used for internal noise generators.
At the macromodel level, representing this frequency
dependence perfectly requires computations with the original
system, thus defeating the purpose of macromodeling. Instead,
it is preferable to find approximate, but computationally
inexpensive, forms of this frequency dependence. Such a
capability has already been obtained for LTI systems with
stationary noise [13], [14]. In this section, we sketch the
extension to cyclostationary noise in LTV systems, useful
for capturing phenomena such as frequency-translation and
mixing of noise.